Interweaving the Numerical Kinematic Symmetry Principles in School and Introductory University Physics Courses

S S symmetry

Brief Report Interweaving the Numerical Kinematic Symmetry Principles in School and Introductory University Physics Courses

Yuval Ben-Abu 1,*, Hezi Yizhaq 2, Haim Eshach 3 and Ira Wolfson 4 1 Department of Physics and Project Unit, Sapir Academic College, Sderot, Hof Ashkelon 79165, Israel 2 Department of Solar Energy and Environmental Physics, Jacob Blaustein Institute for Desert Research, Ben-Gurion University of Negev, P.O.B. 653, Beer-Sheva 8410501, Israel; [email protected] 3 Department of Science and Technology Education, Ben-Gurion University of Negev, P.O.B. 653, Beer-Sheva 8410501, Israel; [email protected] 4 Department of physics, Ben Gurion University of the Negev, 84990, Israel; [email protected] * Correspondence: [email protected]

Received: 18 December 2018; Accepted: 22 January 2019; Published: 29 January 2019

Abstract: The “super-gun” class of weaponry has been around for a long time. However, its unusual physics is largely ignored to this day in mainstream physics. We study an example of such a “super gun”, the “Paris gun”. We first look into the historic accounts of the firing distance of such a gun and try to reconcile it with our physical understanding of ballistics. We do this by looking into the drag component in the equations of motion for ballistic movement, which is usually neglected. The drag component of the equations of motion is the main reason for symmetry breaking in ballistics. We study ballistics for several air density profiles and discuss the results. We then proceed to look into the effects of muzzle velocity as well as mass and ground temperature on the optimal firing angle and firing range. We find that, even in the simplest case of fixed air density, the effects of including drag are far reaching. We also determine that in the “sensible” range of projectile mass, the muzzle velocity is the most important factor in determining the maximal firing range. We have found that even the simplest of complications that include air density, shifts the optimal angle from the schoolbook’s 45-degree angle, ground temperature plays a major role. While the optimal angle changes by a mere two degrees in response to a huge change in ground temperature, the maximal distance is largely affected. Muzzle velocity is perhaps the most influential variable when working within a sensible projectile mass range. In the current essay, this principle is described and examples are provided where students can apply them. For each problem, we provide both the force consideration solution approach and the energy consideration solution approach.

Keywords: numerical analysis; super gun; Paris gun; equations of motion; ballistic movement; muzzle velocity; ground temperature; air density; 45-degree angle

1. Introduction One late-March morning in 1918, a high-pitched whistle pierced the streets of Paris, followed promptly by an explosion. This came as a surprise to many Parisians, literally “out of the blue” [1]. While they were used by that stage of World War I to explosions following the thunders of artillery, or the rickety humming of a biplane on a bomb run, this shelling was relatively stealthy, having little to no forewarning in the way of sound effects. This led to the widespread belief among the shelled citizenry that they were targeted by a high-ﬂying Zeppelin air-ship [1,2]. However, these projectiles were shot some 120 km away by “The Paris gun” (also referred to as “Big Bertha”) [1,2], a behemoth of artillery setup deployed on a hill near Crépy, a town under German

Symmetry 2019, 11, 148; doi:10.3390/sym11020148 www.mdpi.com/journal/symmetry Symmetry 2019, 11, 148 2 of 11 occupation since the German offensive of 1914, all through the war until the final stages in the fall of 1918. This piece of artillery was a modern member of the old and respected family of Superguns: Guns of unusual characteristics, be it a huge bore, a much heavier payload, extremely high muzzle velocity, etc. [2]. 351 shells hit Paris, killing 256 and wounding 620. However, this did not affect the course of the war. What was the discovery that allowed the Germans to increase the range of shells four times than the best contemporary guns? This discovery was accidental [2,3]. The German artillerymen noticed that when the cannon muzzle was tilted at a large angle, the shells traveled far more than expected— 40 km instead of 20 km. When the shell was fired, the high-altitude trajectory reached a thin layer of different atmospheric resistance, before it hit the ground again. This formidable cannon was a strategic rather than a tactical weapon, it was intended to sow fear and panic in the heart of Parisians. The accuracy of the cannon was quite low, making it possible to attack Paris, but with little precision [1–3]. Physics textbooks would have us believe that the optimal firing angle for achieving maximal distance is 45 degrees. This is due to the decoupling of the horizontal and vertical components of the projectile movement neglecting air resistance [4]. In practice, when considering the air drag, the projectile trajectory is much more complex, spawning a rich branch of physics and engineering called exterior ballistics [4,5]. This branch deals with the movement of projectiles after leaving the barrel, and accounts for the variety of forces exerted on and dynamics affecting the shell or shot. For instance, a shell usually moves through a spiral or coil etched or drilled in the gun bore to give it a spiraling movement [6,7]. This is done (among other reasons) to average out the irregularities of the shell along the longitudinal dimension and stabilize its flight. There are additional factors, such as the shape of the shell, the angle of attack, wind, and the influence of Coriolis’ force [6–10]. The purpose of this work is to examine the effect of the atmosphere on a long-range shell [4–6,11].

2. Accounting for Air Resistance The naïve ballistic equations state that the horizontal and vertical components of the movement are decoupled, thus we have ( x = x0 + v0 cos αt gt2 (1) y = y0 + v0 sin αt − 2 .

One can set the initial coordinates x0, y0 to vanish, and by a process of substitution get the trajectory equation. gx2 y = xtan(θ) − 2 2 2v0 cos θ which is a manifestly symmetric trajectory, regardless of the ﬁring angle. This symmetry is sourced by the conservation of energy on both axes and the conservation of momentum on the horizontal axis. Both of which are postulated symmetries in physics in general. Classically then, the ﬁring range is given by:

2 cos(θ) sin(θ)v2 R = 0 g

◦ Which has a maxima at θ = 45 and it is strictly quadratic in the muzzle velocity v0. However, when considering air resistance, the picture is somewhat more complicated [1–3,11]. The drag force term, which encapsulates the resistance of the air (or any sufﬁciently non-viscous medium for that matter) to the projectile traversing through it is given in our case by

→ CAρv2 vˆ F = − , (2) d 2 Symmetry 2019, 11, 148 3 of 11

→ whereSymmetryρ is2019 the, 11 air FOR density, PEER REVIEWv is the projectile velocity, A the cross-section of the projectile along its3 movement direction, and C is some scale-less shape dependent coefficient empirically measured, 85 whichindependent is aptly of called the coordinate. the “drag This coefficient” independence and it is may also depend reflected on in the the projectile process of velocity developing [11,12 the]. 86 ThisEuler-Lagrange drag force, isformalism, independent for instance, of location, where as as it isa ru sourcedle kinetic by energy velocity, is a which, function in of phase velocity space, alone, is 87 independentand potential of terms the coordinate. are function This of independencecoordinates. Thus, is also the reflected drag force, in the by process breaking of momentum developing theand 88 Euler-Lagrangeenergy conservation, formalism, actually for instance,breaks the where symmetries as a rule of kinetic the system. energy Fo isr acircular function bodies, of velocity the value alone, of 89 andC is potential0.5 and it termscan be are as much function as 2 of for coordinates. bodies of irregular Thus, the shape. drag In force, fact, the by breakingsquare velocity momentum model and best 90 energydescribes conservation, the motion actually of most breaks of the theglobular symmetries bodies of we the know system. from For everyday circular bodies,life, accounting the value for of air C 91 isresistance. 0.5 and it can be as much as 2 for bodies of irregular shape. In fact, the square velocity model best 92 describesIn general, the motion the air of resistance most of the for globular a body in bodies motion we depends know from on several everyday factors: life, accountingthe direction for of airthe 93 resistance.body's velocity, its size, body's rotation, air characteristics, and the force of the wind. For small bodies 94 movingIn general, at low thespeeds, air resistance the friction for aforce body (drag in motion force) depends is proportional on several to factors:the speed, the and direction for high of ⃗ 95 thevelocities, body’s the velocity, friction its force size, 𝐹 body’s is proportional rotation, air to characteristics, the speed square and and the can force be written of the wind.as above For [11–13]. small 96 bodies moving at low speeds, the friction force (drag force) is proportional to the speed, and for high 97 For this case then, the→ ballistic equations in the x and the y directions are written: velocities, the friction force F d is proportional to the speed square and can be written as above [11–13]. For this case then, the ballistic equations𝑑𝑣 in the𝐶𝐴𝜌|𝑣|x and the 𝑣 y directions are written: =− 𝑑𝑡 2𝑚 ( CAρ|v| v (3) 𝑑𝑣dvx = − 𝐶𝐴𝜌|𝑣|x 𝑣 dt = −𝑔−2m (3) dv𝑑𝑡y CAρ|2𝑚v| vy dt = −g − 2m 98 These equations are coupled by virtue of |𝑣| = 𝑣 +𝑣 , thus even the decoupling between x q 2 2 99 and Thesey is lost equations and the symmetry are coupled between by virtue the of first|v| and= secondvx + v yhalf, thus of eventhe trajectory the decoupling is broken. between Figure x 1 100 andshows y is the lost result and the of symmetrystudying trajectori betweenes the that first account and second for a fixed half of air the density trajectory 𝜌=𝜌 is broken.. It is easy Figure to see1 101 showsthat, even the result in this of simplest studying of trajectories cases, the that optimal account angle for is a fixeddifferent air density than thρe =textbookρ0. It is easy45-degree to see angle that, 102 even[13–15]. in this simplest of cases, the optimal angle is different than the textbook 45-degree angle [13–15].

103 104 FigureFigure 1. 1.Simulation Simulation of of trajectories, trajectories, factoring factoring in in drag drag force force (left (left panel). panel). The The air air density density is is constant constant in in 105 thisthis case. case. On On the the right right we we see see the the maximal maximal ﬁring firing distance distance is is given given by by ﬁring firing in in a a more more shallow shallow angle angle 106 (~33(~33 degrees) degrees) than than the the canonical canonical 45 45 degree degree angle. angle.

107 GermanGerman artillery artillery thus thus found found that that air air friction friction cannot cannot be be neglected neglected in in realistic realistic ballistic ballistic movement movement 108 estimations.estimations. 109 TheThe previousprevious sectionsection discussed discussed thethe ballistic ballistic equations, equations, adjustedadjusted forfor air air resistance. resistance. InIn orderorder toto 110 developdevelop aa densitydensity proﬁleprofile forfor thethe atmosphere atmosphere wewe cancan turn turnﬁrst first to to classical classical mechanics, mechanics,and andthen thento to 111 thermodynamicsthermodynamics [ 13[13–15].–15].

112 2.1. Atmospheric or Barometric—Choose Your Flavor

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Symmetry 2019, 11 FOR PEER REVIEW 4 Atmospheric or Barometric—Choose Your Flavor 113 ClassicalClassical mechanics,mechanics, along along with with the the ideal ideal gas gas law, la providesw, provides us with us awith naïve a treatmentnaïve treatment of the earth’sof the 114 atmosphere.earth's atmosphere. We look We at alook virtual at a box,virtual with box, the wi verticalth the dimensionsvertical dimensions enclosing enclosing the area theS and area height 𝑆 andh, 115 likeheight Figure ℎ, like2: Figure 2:

116 Figure 2. An inﬁnitesimal volume element. 117 Figure 2. An infinitesimal volume element. The pressure gradient times S is the force exerted on the gas enclosed in the cube by neighboring 𝑆 118 cells,The and thepressure gas itself gradient experiences times a total is gravitationalthe force exerted force ofonρVg the, where gas enclosedρ is the density in the of cube the gas, by 𝜌𝑉𝑔 𝜌 119 Vneighboringis the total volumecells, and of the cube,gas itself and experiencesg is earth’s gravitationala total gravitational acceleration force constant. of , Wewhere assume is that the 𝑉 𝑔 120 thedensity cube of of gasthe isgas, in mechanical is the total and volume thermal of equilibrium, the cube, and thus: is earth's gravitational acceleration 121 constant. We assume that the cube of gas is in mechanical and thermal equilibrium, thus: ( → ΣΣ𝐹F⃗ =0= 0 (4)(4) PV𝑃𝑉= NRT = 𝑁𝑅𝑇,, 122 Where T is temperature and V is volume and is constant. Specifying the forces, one gets where T is temperature and V is volume and is constant. Specifying the forces, one gets 𝑆𝑝(ℎ) −𝑝(ℎ+𝑑ℎ)−𝜌𝑚𝑉𝑔=0 (5) S(p(h) − p(h + dh)) − ρmVg = 0 (5) 123 ⇓ 𝑝(ℎ+𝑑ℎ) −𝑝(⇓ℎ) =−𝜌𝑚(𝑑ℎ)𝑔. (6) 124 We now use the connection 𝑃=𝜌𝑅𝑇p(h + dh ) to− findp(h )that= − ρm(dh)g. (6) We now use the connection P = ρRT to𝑑𝜌 ﬁnd that𝜌𝑚𝑔 =− , (7) 𝑑ℎ 𝑅𝑇 dρ ρmg = − , (7) 125 which yields the following solution: dh RT which yields the following solution: 𝜌=𝜌𝑒 . (8) − mgh 126 Notice however that this equation is validρ = onlyρ0e whereRT . the temperature 𝑇 is height independent. (8) 𝑇=𝑇(ℎ) 127 WhereverNotice however, thatwe have this equationto take that is validinto account: only where the temperature T is height independent. Wherever T = T(h), we have to take that𝑑𝑝 into account:𝑅𝑇𝑑𝜌 𝑅𝜌𝑑𝑇 = + . (9) 𝑑ℎ 𝑑ℎ 𝑑ℎ dp RTdρ RρdT 128 Taking the leading order approximation= of 𝑇(ℎ)+ as: . (9) dh dh dh 𝑑𝑇 ( ) ( ) ( ) Taking the leading order𝑇 approximationℎ ≃𝑇 ℎ + of T(hΔ𝑇) as: = 𝑇 ℎ +𝐿Δ𝑇, (10) 𝑑ℎ

129 where 𝐿 is the lapse coefficientdT that starts at the height of ℎ , we get: T(h) ' T(h ) + ∆T = T(h ) + L ∆T, (10) 0 dh 0 h0 𝑅𝑇𝑑𝜌 𝑅𝜌𝑑𝑇 𝑅𝑇𝑑𝜌h0 + = + 𝑅𝜌𝐿 = −𝜌𝑚𝑔 (11) 𝑑ℎ 𝑑ℎ 𝑑ℎ 130 131 ⇓

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where Lh0 is the lapse coefﬁcient that starts at the height of h0, we get:

RTdρ RρdT RTdρ + = + RρL = −ρmg (11) dh dh dh ⇓ Symmetry 2019, 11 FOR PEER REVIEW mg ! 5 1 dρ dlog(ρ) Lh + = = − 0 R (12) ρ dh dh T + L (h − h ) 0 h0 𝑚𝑔0 𝐿 + 1 𝑑𝜌 𝑑𝑙𝑜𝑔 (𝜌)mg 𝑅 log (ρ)== − 1 + = log−(T + L(h − h )) + Ce (12)(13) RL 0 0 𝜌 𝑑ℎ 𝑑ℎ 𝑇 +𝐿(ℎ − ℎ) ⇓ 𝑚𝑔 −( + mg ) ( ) 1( RL ) log 𝜌 =−1+ log𝑇T0 +𝐿 ℎ−ℎ +𝐶 (13) ρ = ρ0 𝑅𝐿 (14) T0 + L(h − h0) 132 ⇓ Accordingly, if we go for the simplest of atmospheric profiles, we get a fixed temperature and an exponential decay of air density as we go up.𝑇 However, experimental evidence indicates that the (14) density profile is better described𝜌=𝜌 by the second equation [12,13]. 𝑇 +𝐿(ℎ−ℎ) As can be seen in Figure3, the density profile as a function of altitude for the exponential 133 description Accordingly, is very close if to we the go first for order the simplest derivation of atmospheric result, i.e., the profiles, so-called we barometric get a fixed equation; temperature this 134 suggestsand an exponential that the ballistic decay results of air density change as very we little go up. when However, we use experimental the barometric evidence equation, indicates rather than that 135 thethe atmosphericdensity profile one is (Tablebetter1 described). Figure2 byalso the contains second the equation linear atmospheric[12,13]. temperature profile [6,7].

136 137 FigureFigure 3.3. TemperatureTemperature andand logarithmic logarithmic atmospheric atmospheric air air density density proﬁle profile as as function function of of height. height. We notice 138 theWe differencesnotice the differences between the between naive atmosphericthe naive atmosp equationheric and equation the more and detailedthe more barometric detailed barometric equation 139 divergeequation around diverge 40 around km above 40 km the above air surface, the air for surface, the initial for the surface initial temperature surface temperature of 288.15 K. of 288.15 K.

140 As can be seen in FigureTable 3, the 1. Specificationsdensity profil fore theas “Parisa function Gun”. of altitude for the exponential 141 descriptionGround is very closeProjectile to the firs Masst order derivationEffective Staticresult, i.e. the so-called barometricProjectile equation; Velocity this Elevation (◦) 142 suggestsTemperature that the (K) ballistic(kg) results change veryDrag littl Coefficiente when we use the barometric equation,(m/s) rather than 143 the288.15 atmospheric one (Table 106 1). Figure 2 also 0.21 contains the linear atmospheric 45 temperature 1640 profile [6,7]. 144 We factored the different profiles for several elevation angle trajectories to study the differences 145 in trajectory due to air density profile effects. The baseline data we used for the simulations are the We factored the different profiles for several elevation angle trajectories to study the differences 146 following parameters: in trajectory due to air density profile effects. The baseline data we used for the simulations are the 147 following parameters: Table 1. Specifications for the"Paris Gun".

Ground Projectile Mass Effective Static Drag Elevation Projectile Velocity Temperature (K) (kg) Coefficient (∘) (m/s) 288.15 106 0.21 45 1640

148 where for each analysis we set most parameters to be constant, and change one, to examine the effect 149 on trajectory or other derived quantities. The simulation code is fairly simple. We use an adaptive 150 step Runga–Kutta differential equation solver to numerically solve the amended equations of motion 151 that include drag force. At each time step, we calculate the air density according to the barometric or

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Where for each analysis we set most parameters to be constant, and change one, to examine the effect on trajectory or other derived quantities. The simulation code is fairly simple. We use an adaptive step Runga–Kutta differential equation solver to numerically solve the amended equations of motion that include drag force. At each time step, we calculate the air density according to the barometric or atmospheric equations and insert this into the drag force expression. By setting the air density to zero, we can also recover the naïve equations of motion. Some granularity in results usually occur, not due to the differential equation solver, as this yields a smooth function. Rather, the granularitySymmetry 2019 is, 11 due FOR to PEER the stepsREVIEW in parameter that are examined. Time and resource considerations limit6 our step size in the examined parameter values. 152 atmosphericWe see that, equations as suggested and insert before, this into the differencethe drag force in maximal expression. firing By distancesetting the yielded air density from to using zero, 153 thewe morecan also realistic recover barometric the naïve equation,equations is of only motion. slightly Some different granularity from in using results the usually more approximated occur, not due 154 exponentialto the differential air density equation profile. solver, Figure as4 this shows yields the a four smooth flavors function. discussed: Rather, Ballistic the granularity motion in vacuum, is due to 155 Ballisticthe steps motion in parameter in a constant that are airexamined. density, Time and and lastly, resource the exponential considerations and limit barometric our step air size density in the 156 profilesexamined [7,8 parameter]. values.

157

158 FigureFigure 4. 4.While While trajectories trajectories in vacuumin vacuum (left panel,(left panel, largest la trajectories)rgest trajectories) reach a reac staggeringh a staggering firing distance firing 159 ofdistance> 250 km of ,> introducing 250 𝑘𝑚, introducing a constant air a densityconstant greatly air density decreases greatly the firingdecreases distance the (leftfiring panel, distance smallest (left 160 dot-dash).panel, smallest A realistic dot-dash). atmospheric A realistic profile atmospheric yields trajectories profile yields that trajectories reach about that 130 reach km (rightabout panel).130 km 161 The(right difference panel). The between difference using between an exponential using an density, exponential and the density, more detailed and the result moreof detailed the barometric result of 162 equationthe barometric is small, equation around is tens small, to a ar fewound hundred tens to meters. a few hundred meters.

163 3. SoWe Which see Anglethat, as Is suggested Best? before, the difference in maximal firing distance yielded from using 164 the moreWe set realistic out to find barometric which angle equation, is preferred, is only using slightly the different specifications from of using the “Paris the more gun”. approximated The optimal 165 angleexponential for firing air distancedensity profile. should Figure no doubt 4 shows depend the onfour the flavors ground discussed: temperature, Ballistic even motion in the in simplestvacuum, 166 exponentialBallistic motion case. in However, a constant using air the density, temperature and lastly, profile the of expone the atmospherential and makesbarometric the temperature air density 167 dependenceprofiles [7,8]. even more pronounced [6–9]. Additionally, the muzzle velocity should play a major role in finding the optimal angle, since presumably higher velocity means the ability to reach low density 168 regions3. So which and traverse angle is there best? with modest energy losses. We first looked at the optimal angle with a 169 groundWe temperature set out to find of 288.15 which K, angle and foundis preferred, the optimal using anglethe specifications to be 52.7◦, with of the a whooping"Paris gun". firing The 170 distanceoptimal ofangle 128 km.for firing Figure distance5 displays should the results no doub of ourt depend numerical on the simulation ground fortemperature, the initial conditionseven in the 171 correspondingsimplest exponential to the “Paris case. Gun”,However, which using are giventhe te inmperature the above profile Table1 .of the atmosphere makes the 172 temperature dependence even more pronounced [6–9]. Additionally, the muzzle velocity should play 173 a major role in finding the optimal angle, since presumably higher velocity means the ability to reach 174 low density regions and traverse there with modest energy losses. We first looked at the optimal 175 angle with a ground temperature of 288.15 K, and found the optimal angle to be 52.7∘ , with a 176 whooping firing distance of 128 km. Figure 5 displays the results of our numerical simulation for the 177 initial conditions corresponding to the "Paris Gun", which are given in the above Table 1.

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178 178 179 Figure 5. Maximum firing distance as a function of the firing angle. This was simulated with a ground 180179 temperatureFigureFigure 5. 5.Maximum ofMaximum 288.15 K, firing firing and using distance distance the as realisticas a a function function atmospheric of of the the firing firing profile. angle. angle. Rather This This wasthan was simulated asimulated favoured with with45∘a anglea ground ground 45◦ ∘ 181180 fortemperature maximumtemperature distance, of of 288.15 288.15 the K, K, optimal and and using using angle the the realisticis realistic 52.7∘ atmospheric. atmospheric profile. profile. Rather Rather thanthan aa favouredfavoured45 angleangle 181 forfor maximum maximum distance, distance, the the optimal optimal angle angle is is 52.7 52.7◦. ∘ . 182 Ground temperature should be a major factor in firing distance, this is due to the effects of 182 Ground temperature should be a major factor in firing distance, this is due to the effects of 183 temperatureGround on densitytemperature via the should ideal gasbe a law major [13,14] factor. However, in firing wedistance, have not this studied is due theto theeffects effects of of 183 temperature on density via the ideal gas law [13,14]. However, we have not studied the effects of 184 groundtemperature temperature on density on atmospheric via the ideal strata gas and law the [13,14] "breathing". However, effect we of have strata not is wellstudied documented. the effects of 184 ground temperature on atmospheric strata and the “breathing” effect of strata is well documented. 185 We groundhave found temperature (Figure 6) on that atmospheric indeed the strata temperature and the plays "breathing" a key role, effect and of that stra theta is optimal well documented. angle is 185 We have found (Figure6) that indeed the temperature plays a key role, and that the optimal angle is 186 approximatelyWe have found linear (Figure to ground 6) that temperature. indeed the temperature plays a key role, and that the optimal angle is 186 approximatelyapproximately linear linear to to ground ground temperature. temperature.

187 187 − 188 FigureFigure 6. Optimal 6. Optimal angle angle vs. vs. Ground Ground temperature temperature sugge suggestssts a a linear dependence. ThisThis isis duedue to toρ 𝜌∝∝ T 1 𝜌∝ 189188 𝑇and Figureand the the linear6. linear Optimal approximations approximations angle vs. Ground we we have have temperature made. made. The The maximal sugge maximalsts height a heightlinear is also isdependence. also linear linear in temperature inThis temperature is due andto is 𝑇 190189 andinversely is inverselyand correlatedthe correlatedlinear approximations with with the airthe density air densitywe have at the at made. highestthe highest The part maximal part of the of trajectory. theheight trajectory. is also linear in temperature 190 and is inversely correlated with the air density at the highest part of the trajectory. 191 In Inorder order to toexamine examine the the effects effects of ofcrossing crossing differ differentent strata strata on onthe the optimal optimal angle, angle, a more a more direct direct 191 192 approachapproachIn was order was called calledto examine for. for. We We thehence hence effects studied studied of crossing the the effect effects differs of ofentmuzzle muzzle strata velocity on velocity the onoptimal on the the optimal angle, optimal angle.a more angle. The direct The 192 193 assumptionassumptionapproach is, was is,when when called the the for.velocity velocity We henceis issufficient sufﬁcient studied high, high,the theeffect the projectile projectiles of muzzle will will crossvelocity cross over over on to tothe a a different differentoptimal angle.regime The of 193 194 of airairassumption densitydensity [[8,9].8 ,is,9]. when InIn thatthat the case,case, velocity thethe optimal optimalis sufficient angle angl high,e should should the not projectilenot change change will a a lot crosslot until until over a crossa tocross a overdifferent over to to the regimethe next 194 195 nextregimeof regime air density takes takes place. [8,9]. place. Figure In Figure that7 shows case, 7 shows the just optimal that;just that; we angl see we ane seeshould approximate an approximate not change quadratic aquadratic lot behavioruntil behaviora cross of the over angleof the to asthe 195 196 angleanext function as regimea function of muzzletakes of place.muzzle velocity, Figure velocity, until 7 shows some until threshold justsome that; threshold iswe crossed. see isan crossed. Weapproximate then We see then a quadratic constant-like see a constant-like behavior behavior. of the 196 197 behavior.angle as a function of muzzle velocity, until some threshold is crossed. We then see a constant-like 197 behavior.

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2 R = αv0 + βv0 + γ, (15) where α, β, γ are all positive coefﬁcients. They have the combined effect of shifting the quadratic functionSymmetry from 2019 being, 11 FOR symmetric PEER REVIEW around the vertical axis. 8

198 199 Figure 7. also shows the dependence of the firing range on the muzzle velocity. It should not surprise 200 us, that there is a quadratic relation between the muzzle velocity and the firing range. However, 201 whereas in the ballistic case with no drag, the relation is purely quadratic, i.e. of the form R=αv, 202 where α is some coefficient, the drag term introduces a shift symmetry. The range equation is now 198 203 given by: 199 FigureFigure 7. Also 7. also shows shows the the dependence dependence of the the firing firing range range on on the the muzzle muzzle velocity. velocity. It should It should not surprise not surprise 200 us, that there is a quadratic relation between the muzzle velocity and the firing range. However, us, that there is a quadratic relation between the muzzle velocity and the firing range. However, 𝑅=𝛼𝑣 +𝛽𝑣 +𝛾, (15) 201 whereas in the ballistic case with no drag, the relation is purely quadratic, i.e. of the form R=αv, 2 whereas in the ballistic case with no drag, the relation is purely quadratic, i.e., of the form R = αv0, 202 where α is some coefficient, the drag term introduces a shift symmetry. The range equation is now 204 wherewhereα 𝛼,is some 𝛽, 𝛾 are coefficient, all positive the coefficients. drag term introducesThey have ath shifte combined symmetry. effect The of range shifting equation the quadratic is now 203 given by: 205 givenfunction by: from being symmetric around the vertical axis. 206 The projectile mass should also play some role, as it affects the energy balance. More mass means 207 Thesmaller projectile in-flight mass deceleration, should alsorelatively𝑅=𝛼𝑣 play less some +𝛽𝑣 energy role,+𝛾, loss as itto affectsdrag and the thus energy should balance. result (15)in More better mass 208 distance. We would expect then, that the heavier the projectile, the closer we get to reverting to the 204means where smaller 𝛼, in-flight𝛽, 𝛾 are all deceleration,positive coefficients. relatively They less have energy the combined loss to drag effect and of thusshifting should the quadratic result in better 209 ideal 45 degree angle optimal trajectory [4,6–8]. This is evident in Figure 8, where the simulated data 205distance. function We from would being expect symmetric then, around that the the heavier vertical theaxis. projectile, the closer we get to reverting to the 210 is well fitted by a +𝑐 function, with the constant being 45∘, as suspected. It is easy to make the 206ideal 45 degreeThe projectile angle mass optimal should trajectory also play [4 ,some6–8]. role, This as is it evident affects the in energy Figure 8balance., where More the simulated mass means data is 211 207well smaller fittedmistake by in-flight of a 1approximating+ cdeceleration,function, the with relatively results the constant by less a quadraenergy beingtic loss45 function,◦ ,toas drag suspected. however and thus Itthis isshould easymisses toresult the make highin thebetter mass mistake 212 limit. x 208of approximatingdistance. We would the results expect bythen, a quadraticthat the heavier function, the projectile, however the this closer misses we theget highto reverting mass limit. to the 209 ideal 45 degree angle optimal trajectory [4,6–8]. This is evident in Figure 8, where the simulated data 210 is well fitted by a +𝑐 function, with the constant being 45∘, as suspected. It is easy to make the 211 mistake of approximating the results by a quadratic function, however this misses the high mass 212 limit.

213 Figure 8. Simulated data of optimal angle vs. projectile mass suggests that in the high mass limit the air density becomes negligible. Probing the 50–150 kg range alone would have resulted in a different apparent quadratic ﬁt. However, that ﬁt misses the high mass limit. 213

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214 Figure 8. Simulated data of optimal angle vs. projectile mass suggests that in the high mass limit the 215 air density becomes negligible. Probing the 50–150 kg range alone would have resulted in a different 216 apparent quadratic fit. However, that fit misses the high mass limit.

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Yet another interesting aspect of this problem is the energy leakage as a function of projectile mass. As hinted before, inspecting the overall energy loss over the ballistic trajectory should lead to Symmetrya better 2019 understanding, 11 FOR PEER REVIEW of the mass dependence. Figure9 presents the optimal angle vs. mass along9 with the percentage lot of initial energy. An in-depth analysis of the impact velocity shows a good fit is 214 presentedFigure by8. Simulated an exponential data of fit optimal where angle the argumentvs. projectile is themass square suggests root that of in the the mass. high mass This limit makes the sense 215 air density becomes negligible. Probing the 50–150 kg range alone would have resulted in a different due to the drag being proportional to the velocity squared, whereas the energy term is linear with 216 apparent quadratic fit. However, that fit misses the high mass limit. respect to mass. Figure 10 shows these, as well as the relative error between fit and simulated results.

217 218 Figure 9. A wider view of the optimal angle as a function of mass, along with the percentage of energy 219 loss during flight. This graph reveals that, indeed, the energy loss trends to zero where the large mass 220 limit is taken.

221 Yet another interesting aspect of this problem is the energy leakage as a function of projectile 222 mass. As hinted before, inspecting the overall energy loss over the ballistic trajectory should lead to 223 a better understanding of the mass dependence. Figure 9 presents the optimal angle vs. mass along 224 with the percentage lot of initial energy. An in-depth analysis of the impact velocity shows a good fit 225 is presented by an exponential fit where the argument is the square root of the mass. This makes 217226 sense due to the drag being proportional to the velocity squared, whereas the energy term is linear 218227 withFigureFigure respect 9. 9. AA to wider wider mass. view view Figure of of the the 10 optimal optimal shows angle angle these, as as a aas function function well as of of the mass, mass, relative along along witherror with the the between percentage percentage fit ofand of energy energy simulated 219228 results.lossloss during during flight. ﬂight. This This graph graph reveals reveals that, that, indeed, indeed, the the energy energy loss loss trends trends to to zero zero where where the the large large mass mass 220 limitlimit is is taken. taken.

229 230 FigureFigure 10. 10.A A study study of of the the impact impact velocity velocity suggests suggests the the impact impa velocityct velocity goes goes to the to initialthe initial velocity velocity at the at 231 largethe large mass mass limit. limit. An exponential An exponential function function is apparently is apparently a good a ﬁt, good with fit, a with square-root a square-root of the mass of the as mass the 232 argument.as the argument. The right The panels right showpanels the show residual the residual error (upper error right),(upperand right), the and relative the relative error (lower error right).(lower 233 Theseright). show These the show numerical the numerical errors are errors of the are order of the of or ~2%der of or ~2% less or where less where the function the function is well is ﬁtted. well fitted.

234 AtAt this this point point it mightit might be be beneficial beneficial to mentionedto mentioned yet anotheryet another member member of the of illustrious the illustrious “super “super gun” 235 family,gun” family, the ‘Schwerer the ‘Schwerer Gustav’ Gustav’ (Eng. ‘Heavy (Eng. Gustav’),‘Heavy Gu whichstav’), was which more was than more five than times five heavier times than heavier the 256 ton ‘Paris Gun’. The engineers of the Heavy Gustav have basically made the projectiles so heavy that drag became an almost non-issue. Figure 11 compares the two super guns in terms of optimal 229 degree and firing range, we can immediately see that the Heavy Gustav was not as long-ranged as ◦ 230 the ParisFigure Gun, 10. A and study the of optimal the impact firing velocity angle suggests is closer the to theimpa textbookct velocity45 goes. However, to the initial due velocity to the heavierat 231 the large mass limit. An exponential function is apparently a good fit, with a square-root of the mass 232 as the argument. The right panels show the residual error (upper right), and the relative error (lower 233 right). These show the numerical errors are of the order of ~2% or less where the function is well fitted.

234 At this point it might be beneficial to mentioned yet another member of the illustrious “super 235 gun” family, the ‘Schwerer Gustav’ (Eng. ‘Heavy Gustav’), which was more than five times heavier

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236 than the 256 ton ‘Paris Gun’. The engineers of the Heavy Gustav have basically made the projectiles

237 Symmetryso heavy2019 that, 11 drag, 148 became an almost non-issue. Figure 11 compares the two super guns in terms10 of 11of 238 optimal degree and firing range, we can immediately see that the Heavy Gustav was not as long- 239 ranged as the Paris Gun, and the optimal firing angle is closer to the textbook 45∘. However, due to 240 payloadthe heavier (45–67 payload times heavier),(45–67 times drag andheavier), other drag effects and should other be effects greatly should reduced. be Consequently, greatly reduced. the 241 accuracyConsequently, should the be accuracy greatly increased. should be greatly increased.

242 243 Figure 11.11. TheThe ParisParis Gun, Gun, and and the the Heavy Heavy Gustav. Gustav. The The Gustav, Gustav, a much a much heavier heavier super super gun, ﬁringgun, afiring much a 244 heaviermuch heavier shell has shell reduced has reduced ﬁring range,firing range, but due but to due being to lessbeing susceptible less susceptible to atmospheric to atmospheric (and other) (and 245 effectsother) effects is more is accurate. more accurate.

246 4. Conclusions 247 The wartimewartime employmentemployment ofof “super-guns”,"super-guns", while while not not a new idea, gives us thethe opportunityopportunity toto 248 delve deeperdeeper into into some some advanced advanced ballistic ballistic ideas. ideas. As anAs example, an example, we looked we looked into the into “Paris the gun”,"Paris which, gun", 249 bywhich, classroom by classroom physics alone,physics should alone, have should reached have areached staggering a staggering firing distance firing of distance ~250 km. of However,~250 km. 250 ifHowever, one takes if theone simplified takes the notionsimplified of fixed notion air of density, fixed air the density, distance the is greatlydistance suppressed, is greatly suppressed, even to the 251 pointeven ofto beingthe point largely of ineffective.being largely A goodineffective. approximation A good toapproximation the actual range to liesthe somewhereactual range in thelies 252 middlesomewhere of the in twothe middle extremes. of the This two is extremes. given by usingThis is the given air by density usingprofile the air function,density profile be it thefunction, more 253 accuratebe it the more so-called accurate barometric so-called equation, barometric or the equa moretion, rudimentary or the more exponential rudimentary profile. exponential This was profile. our 254 firstThis motivation,was our first but motivation, we have lookedbut we deeper have looked into the deeper effects into of projectilethe effects mass, of projectile ground mass, temperature, ground 255 etc.temperature, We claim etc. that We these claim concepts that these can concepts and should can beand incorporated should be incorporated in the basic in toolbox the basic of ballistictoolbox 256 physics,of ballistic and physics, not relegated and not off-handedly relegated off-handedly to the obscure to regionsthe obscure of ballistic regions engineering. of ballistic Someengineering. effects 257 areSome overlooked effects are in overlooked this analysis, in suchthis analysis, as the dependence such as the of dependence the drag coefficient of the drag on shellcoefficient velocity. on Also,shell 258 therevelocity. are Also, atmospheric there are effects atmospheric not discussed effects herein not di thatscussed may herein have athat large may effect have on a trajectories large effect and on 259 hencetrajectories optimal and firing hence angles, optimal etc.We firing have angles, found thatetc. evenWe thehave simplest found of that complications even the thatsimplest include of 260 aircomplications density, shifts that the include optimal air angle density, from shifts the schoolbook’sthe optimal angle 45-degree from the angle, schoolbook's ground temperature 45-degree angle, plays 261 aground major role.temperature While the plays optimal a major angle role. changes While by the a mereoptimal two angle degrees changes in response by a mere to a hugetwo degrees change in 262 groundresponse temperature, to a huge change the maximal in ground distance temperature, is largely the affected. maximal Muzzle distance velocity is largely is perhaps affected. the Muzzle most 263 influentialvelocity is variable,perhaps whenthe most working influential within variable, a sensible when projectile working mass within range. a Finally, sensible the projectile optimal anglemass 264 behavesrange. Finally, as expected the optimal in the angle large behaves mass limit, as expected where energy in thelosses large mass due to limit, drag where become energy negligible losses anddue 265 theto drag optimal become angle negligible converges and to the a 45 optimal degree angle.angle converges to a 45 degree angle.

266 AuthorAuthor Contributions:Contributions: Conceptualization, Y.B.A. andand I.W.; methodology, I.W. and Y.B.A.; software,software, I.W. andand 267 Y.B.A.;Y.B.A.; validation,validation, Y.B.A.Y.B.A. FormalFormal analysis,analysis, Y.B.A. and I.W.; investigation, Y.B.A.; resources, Y.B.A.; data curation, 268 Y.B.A.;Y.B.A.; writing—Y.B.A.;writing—Y.B.A.; writing—review writing—review and and editing, editing, Y.B.A.; Y.B.A. visualization,; visualization, Y.B.A. H.Y., Y.B.A. H.E. H.Y., and I.W.;H.E. supervision, and I.W.; Y.B.A.; project administration, Y.B.A.; funding acquisition, Y.B.A. 269 supervision, Y.B. A.; project administration, Y.B.A..; funding acquisition, Y.B.A Conﬂicts of Interest: The authors declare no conﬂict of interest. 270 Conflicts of Interest: The authors declare no conflict of interest.

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